Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with $\mathbb{Z}_m$ symmetry and find the classification of SPT phases depends on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.Comment: published versio

Long-Range triplet Josephson Current Modulated by the Interface Magnetization Texture

We have investigated the Josephson coupling between two s-wave superconductors separated by the ferromagnetic trilayers with noncollinear magnetization. We find that the long-range triplet critical current will oscillate with the strength of the exchange field and the thickness of the interface layer, when the interface magnetizations are orthogonal to the central magnetization. This feature is induced by the spatial oscillations of the spin-triplet state |\uparrow\downarrow>+|\downarrow\uparrow> in the interface layer. Moreover, the critical current can exhibit a characteristic nonmonotonic behavior, when the misalignment angle between interface magnetization and central ferromagnet increases from 0 to \pi/2. This peculiar behavior will take place under the condition that the original state of the junction with the parallel magnetizations is the \pi state